Multivariate time series — VAR and VECM

Vector autoregressions (VARs) extend AR models to multiple series. They are the lingua franca of empirical macroeconomics — Sims's 1980 Nobel-cited paper introduced them as an atheoretical alternative to structural macro models. In finance, VARs underpin macro factor modelling, multivariate risk forecasting, and impulse-response analysis.

VAR(p) definition

Y_t = c + A₁ Y_{t-1} + A₂ Y_{t-2} + ... + A_p Y_{t-p} + u_t
Y_t ∈ Rⁿ,  A_i ∈ Rⁿˣⁿ,  u_t ~ WN(0, Σ_u)

Each component of Y_t is regressed on the recent past of all components. With n = 3 and p = 4, you have 3 equations, each with 1 + 3·4 = 13 parameters = 39 params plus the n(n+1)/2 = 6 unique entries of Σ_u.

Estimation

Each equation can be estimated by OLS, equation by equation, because all equations have the same right-hand side variables (same regressors). This is a special case of SUR (seemingly unrelated regressions) where OLS = GLS.

Lag selection

Same AIC/BIC trade-off as in univariate ARMA. BIC heavily penalises additional lags for n > 1; in practice, financial monthly VARs rarely have p > 4, and daily VARs rarely p > 5.

Granger causality

X Granger-causes Y if past X helps predict Y beyond the predictive power of past Y alone. Operationally: in a VAR for (X, Y), test the joint significance of the X-lag coefficients in the Y equation. Granger causality is not the same as economic causality — it's predictability — but it remains a useful first-pass diagnostic.

Impulse-response functions

An IRF traces the response of every variable in the system to a one-time shock in one variable. Computed by inverting the VAR to the MA(∞) representation. The dominant decay rate is governed by the largest eigenvalue of the companion matrix.

VECM — VAR for cointegrated systems

ΔY_t = c + Π Y_{t-1} + Σ Γ_i ΔY_{t-i} + u_t

When components of Y are individually I(1) but cointegrated, the VAR in levels is misspecified. The vector error-correction model splits dynamics into long-run (Π Y_{t-1}, where rank(Π) = number of cointegrating relations) and short-run (Γ_i ΔY_{t-i}) components. Estimated jointly by Johansen ML.

Forecast error variance decomposition

Given a structural identification, the variance of the h-step-ahead forecast error of variable i can be decomposed into shares attributable to shocks in each variable. Useful for asking 'how much of equity volatility is driven by monetary policy shocks vs term premia shocks'.

VARs in finance

• Equity-bond-currency dynamics: cross-asset risk forecasting and risk-on/risk-off identification.
• Macro factor models for sovereign credit: VAR on growth, inflation, FX, debt, with credit spreads.
• Yield curve macro models: VAR on a few yield-curve factors plus macro inputs (Ang-Piazzesi 2003).
• Risk premia decomposition: separating macro from financial drivers via SVAR.

Curse of dimensionality

VAR(p) on n variables has n²p slope parameters. For n = 20, p = 4: 1600 parameters. Estimating that on monthly data (~360 obs) is over-fitting. Bayesian VARs (Minnesota prior, Litterman 1986) and LASSO-VARs shrink coefficients toward sensible defaults — the modern way to handle larger systems.